Geodesic
Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism
Journal of the European Mathematical Society, Tome 19 (2017) no. 9, pp. 2811-2893

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We study the derived representation scheme DRepn​(A) parametrizing the n-dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRepn​(A) is isomorphic to the Chevalley–Eilenberg homology of the current Lie coalgebra gln∗​(Cˉ) defined over a Koszul dual coalgebra of A. This gives a conceptual explanation to some of the main results of [BKR] and [BR], relating them (via Koszul duality) to classical theorems on (co)homology of current Lie algebras gln​(A) . We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we define the derived affine scheme DRepg​(a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRep_g(a) is isomorphic to the Chevalley–Eilenberg homology of the Lie coalgebra g∗(Cˉ), where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a canonical DG algebra map Φg​(a):DRepg​(a)G→DReph​(a)W , relating the G-invariant part of representation homology of a Lie algebra a in g to the W-invariant part of representation homology of a in a Cartan subalgebra of g. We call this map the derived Harish-Chandra homomorphism as it is a natural homological extension of the classical Harish-Chandra restriction map.
DOI : 10.4171/jems/729
Classification : 17-XX, 16-XX, 18-XX, 53-XX
Keywords: Derived representation scheme, Lie algebra cohomology, Chevalley–Eilenberg complex, Harish-Chandra homomorphism, Koszul duality, Macdonald identity 
@article{JEMS_2017_19_9_a5,
     author = {Yuri Berest and Giovanni Felder and Sasha Patotski and Ajay C. Ramadoss and Thomas Willwacher},
     title = {Representation homology, {Lie} algebra cohomology and the derived {Harish-Chandra} homomorphism},
     journal = {Journal of the European Mathematical Society},
     pages = {2811--2893},
     publisher = {mathdoc},
     volume = {19},
     number = {9},
     year = {2017},
     doi = {10.4171/jems/729},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/729/}
}
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Yuri Berest; Giovanni Felder; Sasha Patotski; Ajay C. Ramadoss; Thomas Willwacher. Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism. Journal of the European Mathematical Society, Tome 19 (2017) no. 9, pp. 2811-2893. doi: 10.4171/jems/729

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